Slide-rule.



G. D. ALLAN.

SLIDE RULE.

APPLICATION FILED MAR. 19, 1914.

Patented Sept. 15, 1914.

Neem...

UNITED strappi @guur carica.

CHARLES D. ALLAN, @E CHICAGO, ILLINOIS.

SLIDE-RULE.

To all whom it may concern Be it known that I, CHARLES D. Aman, a citizen of the United States, and a resident of Chicago, in the county of Cook and State of Illinois, have invented a certain new and useful Improvement in Slide- Rules, of which the following is a speciiication.

My invention relates to engineers slide-- rules,having the general characteristics of construction of the Mannheim rules, and consisting of two fixed bars, joined together by a cross piece,and a movable slide interposed between them, the faces of the bars and slide, on which the various logarithmic scales are arranged being in a'common plane, and my invention relates more particularly to an improved and special arra-ngement of the said logarithmic scales, by meansv of which equations showing the mathematical relationship between volume lof flow, loss of pressure due to friction,

and diameter of conduit and between volume of flow, loss of pressure Adue to friction and velocity of flow, of luids flowing in conduits of circular cross section may be ,Solved, and also the application of these equations tothe determination of the coinbinations 'of standard commercial sizes ofV pipe best suited to the purpose in hand.

In the designing of systems of heating and ventilation, to which my invention is particularly, though by no means exclu# sively,.adapted, I' know of no mechanical device by which the solution of equations of the -above described character can be relationships between-volume, loss of pressure due to friction, diameter of/pipe and velocity of flow, are laboriously tabulated or chartedthe data so arranged/covering only a limited range, by reason of the labor involved in its computation and arrangement for use.

My invention, therefore, has for its object t0 provide va rule by means of which problems in this field, such for examples as those involved in vthe iiow-of-steam and water in standard steel and wroughtiron pipe, maybe readily andaccurately solved over a large range of variation.-

My invent-ion has for its further object to determine the loss of pressure due-to friction of .a constant volumevof flow through equal lengths of consecutive sizes of stand- Specication of Letters Patent. Application led March 19, 19M. Serial No. 825,742.

ard commercial pipe. For example, if in Patented Sept. 15, 1914.

a system oi hot water heating, the ,supply l line to a radiator is made of one size oi pipe, and the return connection of the next smaller consecutive size, each being of equal length, and the volume of flow in each equal, a frictional effect would be obtained o which would be somewhere between the frictional ei'liect 'which would be obtained if both supply and return were' of the samel size as the larger pipe, on the one hand, or of the smaller pipe, on the other hand; or the frictional eiiect would be equal to that obtained, if both supply and return were of the same size, but of a diameter somewhere between that of the large and small pipes.

The form of slide-rule in most common use, generally known as the Mannheim rule, is practically limited to the solution of equations which take the general form" l b a: b :r2 in which a, b and c are the known quantities and the unknown quantity, which it is desired to find. This form of equation, however, is not suitable for determining the relative values of the different factors involved in the How of fluids in circular conduits, as will be hereinafter explained.

'In any conduit of circular cross section the volume of iiow per unit of time, may be represented by the equation of general form,

l@lV-:volume of flow in units of bulk measn sary to balance the loss of pressure due to friction in a pipe of unit length may be represented by the equation of general form in which.

length.'

the relatinship of h to d back to direct inozvelocity of iow in units of lineal measas between-volume, `friction head and diameter, and

l h2 Wy (4) C;=T+

as between volume, friction head and velocity, in which C3 and C4 are constants and W, k, d and u and the exponents X and y, have the same value as in equations (l) and (2). lIn the older, and until recently, the most commonly accepted form of hydraulic'formulae, the value given for X and y are :0:9 and y'zl, which values, if applied to equation (2), would show that the friction head loss varies directly as the square of the velocity and inversely as the diameter, and lif applied to equation (3) would show that the friction head loss would vary directly as the square of the volume and inversely as the fifth power of the diameter, a discrepancy in its relation to the diameter,`that is only apparent and not actual, as WV is a composite quantity having d2 as a direct factor as per equation (l) so that W2 has d* as a direct factor, bringing verse ratio. Recent and reliable data, deducted from the blocking out of the results of actual experimental work on logarithmic paper, in which the slope of the lines of diameter and velocity determine the values of X and y,. or the still more precise method of least squares, show that the actual values of X and y are fractional, and that X is somewhat less than 2 and y somewhat more than l, 2x4-y being, however, still equal to 5, or very nearlyl so. Consequently, if the values 2 andl were assumed, the resulting equations would be only approximately accurate, "and the coefficient would hold good over only a limited range of pipe sizes, while if the true fractional values are taken, the equation will hold true for all sizes of ipe, for fluid of the same viscosity and pipe, the inside surface -of which is of the same roughness.

Referring aga-in to equations and (4) it is evident that they could be partly solved on the regular Mannheim rule, as the lconstants being known, the relative values of L, Wx, and du in equation (3) and El,

Wy and @my in equation (4) could be determined, but their complete solution necessitates certain auxiliary logarithmic scalesiy means of which X and y roots of x yand Wy and 2X-ky roots of du and invention, but, in order to illustrate its use by a drawing, it has been necessary to give the exponents X and y and the constants C3 and C4 definite numerical, values and to assume a definite system of units of measurement,l as it would obviously be im possible to construct an illustrative drawingl of a device of this character, without a system of scale measurement. )Vith this object in view, I have given these exponents and constants a numerical value that fhas worked out successfully inf practice, in designing systems of hot water heating, omitting, however, the detailed algebraic and arithmetical work in deriving them, as not being essential to the principle involved, and kgiving only the finished equations. Corresponding 1n form to equations (3) as between Volume, loss of pressure due to friction, and diameter, the actual equation used is h WL875 in which W--British thermal unitsper hour given off by the water of circulation. It is a multiple of volume of flow in accordance with the number of heat units which one unit 'of volume loses in making the circuit.

Zz-inside diameter of pipe in inches. hzheight in lineal feet of a column of water that will balance the pressure lost in friction through lineal Y feet of pipe.

C5=a variable whose value changes in accordance with the drop in temperature of the circulating water in passing through the system. For example, if the circulating water drops 30 deg. in making the circuit, it will give off twice as much heat er unit of volume, as measured in l( 6L' Lc andere Corresponding in form to equation (4f), as between volume, loss of pressure due to fric tion and velocity, the actual equation iswater that will balance the pressure lost in friction through 100 lineal eetof pipe.` @62a variable with different values corresponding to diderent drops in temperature of the water of circu-y lation, as in equation (5). Some ci the values o C, for different terny perature drops are as follows For Q0 deg. drop 06:.000004724 :rooooosm 5 :000002846 2.0000022347 :.000001987 C, and C4 are designated as constants in the fundamental equations (3) and (4r) while C, and C., are called variables in equations (5) .and (6). This is strictly true in so each case, for in the fundamental equations (3) and (a) W represents volume only, to

some standard of bulk measurement, While as applied to equations (5) and W rep resents heat carrying capacity, a multiple of ec volume, the ratio of which to volume, de-

pends onthe heat distributed per some unit of volume, so that if C5 and Cf, wereworlred out in detail, it would be found that each was a composite quantity, partly constant, but each a function ot a common variable T, which in this case is the drop in temperature of the circulating water, and in steam work might be the product ot the density by the B. T. Ufper pound weight. Hence in the two equations (5) and (6) five true variables only appear, the friction head loss (it) the B. T. U. penhour (W) the diameter (d) the velocity (n) and the temperature drop (T), of which h, W and T are common to both equations (5) and (6) while d and u each appear in only one of these equations, but have a common eX Jonent.

n the accompanying drawing Figure l CC represents an upper face view of my improved slide-rule. Fig. 2 shows a trans-` verse view taken on line mf-'f/n.

li represents the main piece of my .improved slide-rule which consists of two fixed parallel bars M and N and a cross piece l? r=height in lineal feet of a column of llength of the rule.

connecting' their under faces. The bars M and N are provided at their inner edges with longitudinalgrooves h in which is guided. a slide Q provided at its edges with longitudinal. tongues a which engage the grooves h all substantially in accordance with the well known construction of the Mannheim rules. @n the face of the Xed bar M and. progressingfrom left to right in the. longitudinal direction of the rule are two logarithmic scales A-B and C. By a logarithmic scale ll mean one in which the distance of any number on the scale, from the beginning of the prime division in which it is located, as indicated by'its line of gradnation, represents the mantissa of its logarithm, a prime division including all num bers between any two consecutive even powers of ten`.` Scale C adjoining the movable slide Q, which may be considered the key scale of the rule, has four prime divi-- sions, each one-quarter of the graduated Scale A-B has four complete prime divisions and part of a fifth, and is divided into two parts A and B, the

. part being placed over and parallel to the part B, so that the. range ofcalculations desired may be covered in a rule of a length that is commercially feasible. The distance on scale .l-B representing the mantissa of the logarithm of any number is 1.875 times as great as the distance on the f* key scale C representing the mantissa of the logarithm of the same number, and the relai tive longitudinal position of the two scales is such thatthe zero power of ten, which is unity on each scale, would correspond, if continued to that extent, so that all numbers on scale A--B are situated opposite their 1.875 powers on scale C, and conversely, all numbers onscale (l are opposite their 1.87 5 roots on scale A--B, but all num bers on scale .C must be considered as multiples of di`erent powers of ten relative to part A than to part B. The letters T and M following the numbers on scale A-B indicate thousands and millions respectively. On the face of the liXed bar N are also two logarithmic scales H and ill-K.. Scale H adjoining the movable slide Q and progressing from right to left, opposite to the direction of progression ofthe key scale (l, has two prime divisions, each one-halt of the graduated length of the rule, so that all numbers on scale H will be opposite the re ciprocals or their squares on scale (l and, conversely all numbers on scale C will be opposite the reciprocals of their square roots on scale H. Scale J-K, progressing from left to right, has four primedivisions and part of a fifth, and is divided into two parts J and l as scale A-B is divided, and for the same reason. 0n' scale J-K the distance representing the mantissa of the logarithm of any number is 1.25 times as great as on the key scale'C, so that, the position of unity corresponding on each, numbers on scale JK are opposite their 1.25 powers on scale C and conversely numbers on scale C are opposite their 1.25 roots on scale J-K. Theletters T and M following the numbers on scale .I-K yindicate thousands and millions, respectively. y

On'the face of the movable slide Q, arc threelogarithmic scales D, G and' E'F, all progressing from left to right in the longi tudinal direction of the rule. Scales D and lGr are partial reproductions oi the keyscale C, only the lines of graduation terminatingr a prime division, or even powers of "ten, being indicated. On scale D, in its second prime division from the left end of the rule,

are additional lines ol", graduation, repre.

i' one prime division (1 -to 10) and parts of two others adjoining. Distances corresponding to the mantissa of the logarithm of any number are five times as great as on the key scale C, or its partial reproductions D and G and the relative longitudinal positions are such that the points oi unity would correspond. It is a scale of 5th roots relative to scales D and G. y

In addition to the regular lines of graduation in scale E-F heavier lines of graduation indicate in their proper places on the scale the various commercial sizes of steel and iron pipe, from to 26 O. D., inclusive, located as called for by the mantissa of the logarithms of their inside diameters in inches. Sizes 14 to 26 have, after the figures indicating their size, the letter O. D. indicating outside diameter by which commercial pipe over 12" inside diameter is designated. Also between each two consecutive commercial sizes to 3 inclusive there is shown by a heavy line of graduation an intermediate imaginary size marked I.

Referring to equation (5) it is shown that if the volume (W) of ow remain constant, the loss yof head due to friction (la) varies inversely as the 5th power of the diameter (d). Iis an imaginary additional'size of pipe, of diameter the reciprocal of whose fifth power, is a' mean between the reciprocals' of the fifth powers of the diameter of the two adjacent commercial sizes between lwhich it is situated, and if such a size existed it .would give the same rictional e'lect for a given length as though half its length were of the commercial size next below it and the other half o'l' its length of the commercial size next above it, for the same volurne or" flow.

The method of using my improved sliderule is asfollowsr-For the solution of equations between volume of flow, loss oi' pres- 'sure due to friction and diameter, having the form of equation (5) scales A-B, C, D

and E-F are used. Whatever the position of the slide, the loss of pressure due to friction, in terms of feet head per 100 lineal feet of main, on scale C, will be opposite the graduations, corresponding to the drop in temperature of the water of circulation on scale D, and the B. T. U. (British thermal units) per hour on part A or B of scale A-B will be opposite the theoretical diameter in inches on part E or F of scale E.-I*`- by opposite I mean lying in the .same verl tical line, perpendicularl to the horizontal` axisof the rule. By means of the heavy extra lines of graduation on scale E-F representing the commercial sizes oi pipe, selection of the size that will most nearly meet any given requirements may easily be made, and each additional line of graduation marked I, between each two consecutive commercial sizesfrom t to 3 inclusive, indicate an additional imaginary size, which would have the samefriction head loss for equal volume of How, as a line made up of equal lengths of the two consecutive sizes between which it is located.K These lines of graduation, of intermediate frictional effect, are not carried above 3 as their use in practical designing is mostly confined to making connections to an individual radiator, where the supply pipe is of one size and the return pipe of another and both of equal length, particularly in gravity hot water heating, where the calculations are carried to a fine point of eXactness. For the solution of equations between volume of fiow, -loss of lpressure due to friction and velocity of flow,

aving the form of equation (6) scales J-K, I-I, G 'andE-F are used. Whatever the position of the slide, the loss of pressure duel to friction, in terms of feet head per 100 lineal feet oi main on scale II, will be opposite the constant corresponding to the drop in temperature of the water of circulation on scale G, and the B.'I`. U. per hour on part J or P of scale JK will be opposite the velocity`of flow in feet per second on part E or F of scale lil-F.

. A noticeable feature of my invention is that in the solution of equations ofthe form of equation (5) as between volume of flow, friction head loss and diameter, and of the form of equation (6) as between volume of flow friction'head loss, and velocity of flow,

`the same scale of 5th roots serves Yio instan-ee, and velocities in the latter. This is `so for the reason that, in their fundamental forms as'per equations (3) and (4), the exponent of diameterta) and velocity (e) are the seghe, being equal to `20H-y.

A. few ak/tual examples will serve to still further illustrate the practical operation of my invention.

Example TlVhat would be the diameter and velocity of a flow in a pipe carrying 10,000,000, B. T. U. er hour, on'a basis of a temperature drop o 25 degrees, that `is each pound weight of water flowing delivers 25 B. T. U per hour, the loss of head due to friction being 3. feet per 100 feet? Place 25 on scale D opposite 3 on scale C and opposite 10 M on part A of scale A-B read 6.9 inches diameter on part E of scale E-F. The nearest commercial size is *'v.

Place 25 on scale G opposite 3 on scale H i and opposite 10 M on part K of scale l-.l'

read 7.1 feet per secpnd Velocity on part E of scale lil-F.

Example The carrying capacity of a main is 20 million B. T. U. per hour and in giving up this volume of heat it drops 20 degrees in temperature and its friction head loss is 2.5 feet per 100 feet, what is its diameter and velocity of flow? Place 20 on scale D opposite 2.5 on scale C andropposite 20. on part A of scale A-B read 10 (approximate) diameter on part E of scale E-F-10 would be the nearest commercial size.

Place 20 on scale Gr opposite'2.5 on scale H and opposite 20 M on part K of scale J-K find 8.3 feet per second velocity on part E of scale E-F.

Examplez-What is the heat carrying c..- pacity and velocity of flow of 2l commercial pipe on the basis of a temperature drop of 35 ldegrees and a friction head loss of .45 of a foot per 100l feet?- Place 35 on scale D opposite .45 on scale C and opposite sizet2 on part F of scale E-F read 200,000 B. T. U. per hour on part B of scale A-B.

, Place 35 on scale G opposite .45 on scale H and opposite 200 T on part J of scale VlV-K read 7.2 feet per second velocity on part E of scale E-F Example :`-ln a system of hot water heating, a radiator gives off 10,000 B. T. lU. per hour, what size connections should be made to it that the frictional loss may be at the rate of one-tenth of a foot per 100 lineal feet of main and the drop in temperature 35 degrees? Place 35 on scale D opposite .l on scale C and opposite 10 T on part B of scale AMB read T on part F of E-F lying between the consecutive commercial sizes and l. Consequently if one connection to the radiator should be 1-and the other and each be of equal length, the desired frictional effeet will be obtained.

Example In a system of hot water heating a pipe coil gives off 45,000 B. T. U..per hour, the drop in temperature of the water inpassing through' the coil is 20 degrees, what size connections should. be made to gire a frictional, loss at the rate of four-tenths of afoot per 100 feet lineal feet ofmain? Place 20 on'scale D opposite .4 on scale C and opposite 45 T on part B of scale A-i-B read T on part F of scale E-F located between the consecutive commercial sizes lili and 14. Consequently if one connection to the coil is l-l-f and the other 14 and both are of equal length, the desired frictional effect will be obtained.

ldhile my invention is useful in heating work, particularly in problems relating to the flow of low pressure steam and hot water in pipes, it is not confined to that art, but, is generally applicable to the flow of fluids in conduits of circular cross section, such for example as high pressure steam power piping, air in blow pipe systems for the removal of refuse, or water in municipal or other water supply systems, provided the constants are changed to `suit the system of measurements adapted for time and volume, and the exponents made to conform with the viscosity of the fluid, and roughness of interior pipe surface.

T claim l. A slide-rule consisting of two lixed bars and a movable slide interposed between them, having two independently coacting series of scales, relating to the dow of fluids in conduits, arranged for the solution of two forms of equations in which five different variables appear, three of which variables appear in both forms of equation, and each of the other two variables appear in only one of the forms of equation, but have a common exponent, each of the two fixed bars of the rule having two logarithmic scales graduated to read the values of two of the variables appearing in both forms of equation, and on the slide two scales each graduated to read the values of the third remaining variable common to both forms of equation, and on the slide a third scale, graduated to 'read the values of one of the two variables with common exponent not appearing in both forms of equation, when coacting with the two logarithmic scales on one fixed bar and one of the other scales on the slide, the same graduations on said third scale of the slide serving to read the values of the other of the two variables with common exponent, when coacting with the two logarithmic scales on the other fixed bar and the remaining other scale, on the slide, the graduations on the saidA third scale on the slide in each case representing a different character of quantity measured by a (liderent system of units, substantially as de scribed.

2. A slide-rule consisting of relative movable members having coacting scales one of which comprises lines of graduation indieating consecutive sizes of commercial material and additional lines of graduation indicating sizes of material which Wouid produce a mechanical eeot which is a mean between the same mechanical eect produced bv the two consecutive commercial sizes be- CHARLES D. ALLAN.

Witnesses:

ALEX. D. KING, Jr. A. H. GUNGGOLL. 4 

